Strong-to-Weak Spontaneous Symmetry Breaking in a $(2+1)$D Transverse-Field Ising Model under Decoherence

Abstract

Decoherence in many-body quantum systems can give rise to intrinsically mixed-state phases and phase transitions beyond the pure-state paradigm. Here we study the $(2+1)$D transverse-field Ising model subject to a strongly $\mathbb{Z}_2$-symmetric decoherence channel, with a focus on strong-to-weak spontaneous symmetry breaking (SWSSB). This problem is challenging because the relevant transitions occur in the strong-decoherence regime, beyond the reach of perturbative expansions around the pure-state limit, while conventional quantum Monte Carlo (QMC) methods are hampered by the need to access nonlinear observables and by the sign problem. We overcome these difficulties by developing a QMC algorithm that efficiently evaluates nonlinear Rényi-2 correlators in higher dimensions, complemented by an effective field-theoretic approach. We show that the decohered state realizes a rich mixed-state phase diagram governed by an effective 2D Ashkin-Teller theory. This theory enables analytical predictions for the mixed-state phases and the universality classes of the phase boundaries, all of which are confirmed by large-scale QMC simulations.

Figures (7)

• Figure 1: Graphical representation of the density matrix in the evolution picture, with propagation from the ket to the bra index in the computational basis. (a) Initial state $\rho_0$. (b) State after a single local channel. (c, d) Two contributions $\sigma_1$ and $\sigma_2$ in $\mathcal{E}_{\langle ij\rangle}[\rho_0]=\sigma_1+\sigma_2$ shown in (b). The yellow diamond in (d) denotes a Kronecker tensor enforcing identical spin states on sites $i$ and $j$ along four time directions. (e) State after the full channel. (f) Contracting two copies of $\rho=\mathcal{E}[\rho_0]$ yields $\rho^2$. Further contracting the bra and ket indices gives $\text{Tr}(\rho^2)$, which can be simulated by QMC to evaluate Rényi-2 observables.
• Figure 2: Phase diagram of the decohered ground state of the 2D TFIM under the quantum channel $\mathcal{E}$. Markers denote critical points obtained from QMC simulations associated with the corresponding order parameters. For ground-state preparation, we take $\beta = 2L$, and verify convergence. The strongly symmetric phase is characterized by $C^{(\alpha)}=0$ for $\alpha=0,1,2$; the R2-SWSSB phase by $C^{(2)}\neq 0$ and $C^{(0)}=C^{(1)}=0$; the R2-SSB phase by $C^{(1)},C^{(2)}\neq 0$ with $C^{(0)}=0$; and the ordinary SSB phase by $C^{(\alpha)}\neq 0$ for all $\alpha=0,1,2$.
• Figure 3: Binder ratios $R^{(\alpha)}$ as functions of system size $L$ and tuning parameters $J$ or $p$. (a) At fixed $J=0.1$, the curves cross at $p_c \approx 0.355$. (b) Corresponding data collapse of $R^{(2)}(p,L)$, yielding $\nu \approx 0.998\approx 1$. (c) At fixed $p=0.3$, the crossing occurs at $J_c \approx 0.29$, close to the tricritical point. (d) Data collapse using the expected exponent $\nu = 2/3$, with all curves collapsing near criticality, in agreement with theory.
• Figure 4: An example for the blue critical boundary in Fig. \ref{['fig:phase_diagram']}. At fixed $p=0.6$, we tune $J$ and evaluate the Binder ratio $R^{(1)}$. (a) Crossings of $R^{(1)}(J,L)$ for different system sizes locate the critical point at $J_c \approx 0.27$. (b) Finite-size scaling yields an excellent data collapse with $\nu \approx 0.98$, close to the expected Ising value $\nu=1$.
• Figure 5: Binder ratios $R^{(\alpha)}$ as functions of system size $L$ and tuning parameters $J$ and $p$. Results for $\alpha=0$ are shown in (a, b), and for $\alpha=1$ in (c, d). We fix $J = 0.328474$. (a, c) show $R^{(\alpha)}$ versus the decoherence rate $p$ for different $L$, exhibiting crossings that locate the critical points, while (b, d) show the corresponding data collapse using $\nu= 1.70$. The crossing points are $p_c \approx 0.01$ in (a) and $p_c \approx 0.04$ in (c).